You need a police clearance (report) to apply for a PNG work permit, so why not get this for the few dollars it costs? Without a work permit you are going nowhere. If an employer does not think that they can get you a work permit they will move on to the next applicant.
In some Australian states certain convictions drop dead after so many years - if that time has elapsed you should find that whatever felony you committed is not on the police report.
I see that you are in NZ which may work differently, in which case you could be screwed.